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## G = D4×C42order 336 = 24·3·7

### Direct product of C42 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C42
 Chief series C1 — C2 — C14 — C42 — C2×C42 — D4×C21 — D4×C42
 Lower central C1 — C2 — D4×C42
 Upper central C1 — C2×C42 — D4×C42

Generators and relations for D4×C42
G = < a,b,c | a42=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 140 in 108 conjugacy classes, 76 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C7, C2×C4, D4, C23, C12, C2×C6, C2×C6, C2×C6, C14, C14, C14, C2×D4, C21, C2×C12, C3×D4, C22×C6, C28, C2×C14, C2×C14, C2×C14, C42, C42, C42, C6×D4, C2×C28, C7×D4, C22×C14, C84, C2×C42, C2×C42, C2×C42, D4×C14, C2×C84, D4×C21, C22×C42, D4×C42
Quotients: C1, C2, C3, C22, C6, C7, D4, C23, C2×C6, C14, C2×D4, C21, C3×D4, C22×C6, C2×C14, C42, C6×D4, C7×D4, C22×C14, C2×C42, D4×C14, D4×C21, C22×C42, D4×C42

Smallest permutation representation of D4×C42
On 168 points
Generators in S168
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 84 102 154)(2 43 103 155)(3 44 104 156)(4 45 105 157)(5 46 106 158)(6 47 107 159)(7 48 108 160)(8 49 109 161)(9 50 110 162)(10 51 111 163)(11 52 112 164)(12 53 113 165)(13 54 114 166)(14 55 115 167)(15 56 116 168)(16 57 117 127)(17 58 118 128)(18 59 119 129)(19 60 120 130)(20 61 121 131)(21 62 122 132)(22 63 123 133)(23 64 124 134)(24 65 125 135)(25 66 126 136)(26 67 85 137)(27 68 86 138)(28 69 87 139)(29 70 88 140)(30 71 89 141)(31 72 90 142)(32 73 91 143)(33 74 92 144)(34 75 93 145)(35 76 94 146)(36 77 95 147)(37 78 96 148)(38 79 97 149)(39 80 98 150)(40 81 99 151)(41 82 100 152)(42 83 101 153)
(1 154)(2 155)(3 156)(4 157)(5 158)(6 159)(7 160)(8 161)(9 162)(10 163)(11 164)(12 165)(13 166)(14 167)(15 168)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 133)(23 134)(24 135)(25 136)(26 137)(27 138)(28 139)(29 140)(30 141)(31 142)(32 143)(33 144)(34 145)(35 146)(36 147)(37 148)(38 149)(39 150)(40 151)(41 152)(42 153)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)

G:=sub<Sym(168)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,84,102,154)(2,43,103,155)(3,44,104,156)(4,45,105,157)(5,46,106,158)(6,47,107,159)(7,48,108,160)(8,49,109,161)(9,50,110,162)(10,51,111,163)(11,52,112,164)(12,53,113,165)(13,54,114,166)(14,55,115,167)(15,56,116,168)(16,57,117,127)(17,58,118,128)(18,59,119,129)(19,60,120,130)(20,61,121,131)(21,62,122,132)(22,63,123,133)(23,64,124,134)(24,65,125,135)(25,66,126,136)(26,67,85,137)(27,68,86,138)(28,69,87,139)(29,70,88,140)(30,71,89,141)(31,72,90,142)(32,73,91,143)(33,74,92,144)(34,75,93,145)(35,76,94,146)(36,77,95,147)(37,78,96,148)(38,79,97,149)(39,80,98,150)(40,81,99,151)(41,82,100,152)(42,83,101,153), (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,161)(9,162)(10,163)(11,164)(12,165)(13,166)(14,167)(15,168)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,151)(41,152)(42,153)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,84,102,154)(2,43,103,155)(3,44,104,156)(4,45,105,157)(5,46,106,158)(6,47,107,159)(7,48,108,160)(8,49,109,161)(9,50,110,162)(10,51,111,163)(11,52,112,164)(12,53,113,165)(13,54,114,166)(14,55,115,167)(15,56,116,168)(16,57,117,127)(17,58,118,128)(18,59,119,129)(19,60,120,130)(20,61,121,131)(21,62,122,132)(22,63,123,133)(23,64,124,134)(24,65,125,135)(25,66,126,136)(26,67,85,137)(27,68,86,138)(28,69,87,139)(29,70,88,140)(30,71,89,141)(31,72,90,142)(32,73,91,143)(33,74,92,144)(34,75,93,145)(35,76,94,146)(36,77,95,147)(37,78,96,148)(38,79,97,149)(39,80,98,150)(40,81,99,151)(41,82,100,152)(42,83,101,153), (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,161)(9,162)(10,163)(11,164)(12,165)(13,166)(14,167)(15,168)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,151)(41,152)(42,153)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,84,102,154),(2,43,103,155),(3,44,104,156),(4,45,105,157),(5,46,106,158),(6,47,107,159),(7,48,108,160),(8,49,109,161),(9,50,110,162),(10,51,111,163),(11,52,112,164),(12,53,113,165),(13,54,114,166),(14,55,115,167),(15,56,116,168),(16,57,117,127),(17,58,118,128),(18,59,119,129),(19,60,120,130),(20,61,121,131),(21,62,122,132),(22,63,123,133),(23,64,124,134),(24,65,125,135),(25,66,126,136),(26,67,85,137),(27,68,86,138),(28,69,87,139),(29,70,88,140),(30,71,89,141),(31,72,90,142),(32,73,91,143),(33,74,92,144),(34,75,93,145),(35,76,94,146),(36,77,95,147),(37,78,96,148),(38,79,97,149),(39,80,98,150),(40,81,99,151),(41,82,100,152),(42,83,101,153)], [(1,154),(2,155),(3,156),(4,157),(5,158),(6,159),(7,160),(8,161),(9,162),(10,163),(11,164),(12,165),(13,166),(14,167),(15,168),(16,127),(17,128),(18,129),(19,130),(20,131),(21,132),(22,133),(23,134),(24,135),(25,136),(26,137),(27,138),(28,139),(29,140),(30,141),(31,142),(32,143),(33,144),(34,145),(35,146),(36,147),(37,148),(38,149),(39,150),(40,151),(41,152),(42,153),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,121),(62,122),(63,123),(64,124),(65,125),(66,126),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102)]])

210 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A ··· 6F 6G ··· 6N 7A ··· 7F 12A 12B 12C 12D 14A ··· 14R 14S ··· 14AP 21A ··· 21L 28A ··· 28L 42A ··· 42AJ 42AK ··· 42CF 84A ··· 84X order 1 2 2 2 2 2 2 2 3 3 4 4 6 ··· 6 6 ··· 6 7 ··· 7 12 12 12 12 14 ··· 14 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 42 ··· 42 84 ··· 84 size 1 1 1 1 2 2 2 2 1 1 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

210 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C7 C14 C14 C14 C21 C42 C42 C42 D4 C3×D4 C7×D4 D4×C21 kernel D4×C42 C2×C84 D4×C21 C22×C42 D4×C14 C2×C28 C7×D4 C22×C14 C6×D4 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C42 C14 C6 C2 # reps 1 1 4 2 2 2 8 4 6 6 24 12 12 12 48 24 2 4 12 24

Matrix representation of D4×C42 in GL3(𝔽337) generated by

 336 0 0 0 333 0 0 0 333
,
 336 0 0 0 336 335 0 1 1
,
 1 0 0 0 336 335 0 0 1
G:=sub<GL(3,GF(337))| [336,0,0,0,333,0,0,0,333],[336,0,0,0,336,1,0,335,1],[1,0,0,0,336,0,0,335,1] >;

D4×C42 in GAP, Magma, Sage, TeX

D_4\times C_{42}
% in TeX

G:=Group("D4xC42");
// GroupNames label

G:=SmallGroup(336,205);
// by ID

G=gap.SmallGroup(336,205);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-7,-2,2041]);
// Polycyclic

G:=Group<a,b,c|a^42=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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